8edf

← 7edf

8edf

9edf →

Prime factorization	23
Step size	87.7444¢
Octave	14\8edf (1228.42¢) (→7\4edf)
Semitones (A1:m2)	0:2 (0¢ : 175.5¢)
Consistency limit	2
Distinct consistency limit	2

8 equal divisions of the fifth (8edf, 8ed3/2) is the tuning system that divides the fifth into 8 steps of 87.7444 cents each, making it very nearly 88cET. It is related to the octacot temperament. 8edf corresponds to about 13.6761edo (similar to every third step of 41edo).

Intervals

8edf can be notated either using native uranian (sesquitave) notation, where the notation repeats every period (i.e. just diatonic fifth), or using double sesquitave (Annapolis) notation, where the notation repeats every two periods (i.e. major diatonic ninth). This interprets 8edf as 16ed9/4, resulting in a tuning for the Natural and Harmonic Minor modes of Annapolis[6L 4s]. It can also be notated using tetratonic 4edf-based notation.

The naturals result from a semiwolf generator (~7/6). For sesquitave notation, letters A-E can be used. For double sesquitave notation, Greek numerals 1-10 can be used (Α,Β,Γ,Δ,Ε,Ϛ/Ϝ,Ζ,Η,Θ,Ι).

Scale	Cents	Approximate intervals			Uranian	Diatonic	Notation
degree	value	7-limit	11-limit	19-limit	interval	equivalent	Uranian	Annapolis	Tetratonic notation
0	0	1			unison		A	Α	C
1	87.7444	21/20	22/21	20/19, 19/18	min mos2nd	minor second	A#	Α#	^C, vD
2	175.48875	10/9		21/19	maj mos2nd	major second	B	Β	D
3	263.2331	7/6			perf mos3rd	subminor third	C	Γ	^D, vE
4	350.9775	60/49, 49/40	11/9		aug mos3rd	neutral third	C#	Γ#	E
5	438.7219	9/7			perf mos4th	supermajor third	D	Δ	^E, vF
6	526.46625	27/20		19/14	min mos5th	wolf fourth	D#	Δ#	F
7	614.2106	10/7		27/19	maj mos5th	augmented fourth	E	Ε	^F, vC
8	701.955	3/2			sesquitave	just fifth	A	Ϛ/Ϝ	C
9	789.6994	63/40	11/7	30/19	min mos7th	minor sixth	A#	Ϛ#/Ϝ#	^C, vD
10	877.44375	5/3		63/38	maj mos7th	major sixth	B	Ζ	D
11	965.1881	7/4			perf mos8th	subminor seventh	C	Η	^D, vE
12	1052.9325	90/49, 35/18	11/6		aug mos8th	neutral seventh	C#	Η#	E
13	1140.6769	27/14			perf mos9th	supermajor seventh	D	Θ	^E, vF
14	1228.42125	81/40		57/28	min mos10th	acute octave	D#	Θ#	F
15	1316.1656	15/7		81/38	maj mos10th	minor ninth	E	Ι	^F, vC
16	1403.91	9/4			double sesquitave	major ninth	A	Α	C

Scale workshop link for a keyboard/MIDI playable 8EDF (with graphical uranian scale, A=220Hz)

Scale tree

EDF scales can be approximated in EDOs by subdividing diatonic fifths. If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger EDOs pop up in our continuum.

Generator range: 85.7143 cents (4\7/8 = 1\14) to 90 cents (3\5/8 = 3\40)

Fifth							Cents	Comments
4\7							85.714
						27\47	86.170
					23\40		86.250
						42\73	86.301
				19\33			86.36
						53\92	86.413
					34\59		86.441
						49\85	86.471
			15\26				86.5385
						56\97	86.598
					41\71		86.620
						67\116	86.638
				26\45			86.6	Flattone is in this region
						63\109	86.697
					37\64		86.719
						48\83	86.747
		11\19					86.842
						51\88	86.9318
					40\69		86.9565
						69\119	86.975
				29\50			87.000
						76\131	87.023	Golden meantone (696.2145¢)
					47\81		87.037
						65\112	87.054
			18\31				87.097	Meantone is in this region
						61\105	87.143
					43\74		87.162
						68\117	87.1795
				25\43			87.209
						57\98	87.245
					32\55		87.27
						39\67	87.313
	7\12						87.500
						38\65	87.692
					31\53		87.736	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	87.766	Garibaldi / Cassandra
				24\41			87.805
						65\111	87.837
					41\70		87.857
						58\99	87.87
			17\29				87.931
						61\104	87.981
					44\75		88.000
						71\121	88.0165	Golden neogothic (704.0956¢)
				27\46			88.0435	Neogothic is in this region
						64\109	88.073
					37\63		88.095
						47\80	88.125
		10\17					88.235
						43\73	88.356
					33\56		88.392
						56\95	88.421	The generator closest to a just 5/3 for EDOs less than 1600
				23\39			88.4615
						59\100	88.500
					36\61		88.525
						49\83	88.554
			13\22				88.63	Archy is in this region
						42\71	88.732
					29\49		88.7755
						45\76	88.816
				16\27			88.8
						35\59	88.931
					19\32		89.0625
						22\37	89.189
3\5							90.000

Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

Rank 2 temperaments

MOS scales and temperament listed by generator size and period:

Periods per octave	Generator	Scale pattern	Temperaments
1	1\8	1L5s, 1L6s	(pathological)
1	3\8	3L2s (uranian)	Semiwolf
2	3\8	2L 2s	(augmented-like)

Images

8EDF cheat sheet (sesquitave notation) - table only